Exact solution to the fully connected XY model with Gaussian random fields by the replica method

We solve the fully connected ferromagnetic XY model in a local Gaussian random magnetic field using the replica method. We work out the replica symmetric (RS) solution and prove its exactness by diagonalizing the Hessian around the RS saddle point and showing that all its eigenvalues are non-negative for all values of the temperature T and random field strength sigma. As a consequence, there is no spin-glass phase with replica symmetry breaking in this model. Moreover, we show that the critical line sigma c(T) separating the paramagnetic and ferromagnetic phases is a strictly monotonic decreasing function of T and thus there is no reentrant paramagnetic phase at low temperature. Furthermore, all points on the critical line are second order phase transition critical points, and thus there are no first-order phase transitions for any value of sigma and T. Finally, we show that the calculation we present here can be easily generalized beyond the XY model to the O(nu) model with an arbitrary number nu of spin components. (c) 2023 Elsevier B.V. All rights reserved.